martin larsson bernhard müller*folk.ntnu.no/bmuller/pcfd_12_2-3_paper10.pdf · reference to this...

164 Progress in Computational Fluid Dynamics, Vol. 12, Nos. 2/3, 2012

High-order numerical simulation of fluid-structure

interaction in the human larynx

Martin Larsson

ONERA-DAFE,8 rue des Vertugadins,F-92190 Meudon, FranceE-mail: [email protected]

Bernhard Müller*

Department of Energy and Process Engineering,Norwegian University of Science and Technology (NTNU),Kolbjørn Hejes vei 2, NO-7491 Trondheim, NorwayFax: +47 73593491E-mail: [email protected]*Corresponding author

Abstract: We present a 2D model of the human larynx able to reproduce the self-sustainedoscillating interaction between the airflow and the vocal folds. We describe the interactionof the compressible flow and the elastic structure in an ALE formulation. Our solver usesglobally fourth-order accurate Summation By Parts (SBP) finite difference operators inspace and a fourth-order explicit method in time. We derive Simultaneous ApproximationTerm (SAT) expressions that impose the velocity and traction boundary conditions weakly.We present results from simulations with realistic parameters for human phonation.

Keywords: fluid-structure interaction; high-order finite difference method; phonation.

Reference to this paper should be made as follows: Larsson, M. and Müller, B. (2012)‘High-order numerical simulation of fluid-structure interaction in the human larynx’,Progress in Computational Fluid Dynamics, Vol. 12, Nos. 2/3, pp.164–175.

Biographical notes: M. Larsson did his MSc in Engineering Physics at Uppsala University,Sweden, in 2007. He received his PhD in 2010 under the supervision of B. Müller fromthe Norwegian University of Science and Technology (NTNU) at Trondheim, Norway. Heis presently working on a postdoc position at ONERA in Meudon, France, investigatingaerodynamic stability for low-aspect ratio flapping wings. His research interests includefluid-structure interaction, phonation and numerical and aerodynamic stability.

B. Müller did his Diploma in Mathematics at the University of Cologne, Germany, in 1978.In 1981, he obtained the VKI-Diploma in Aeronautics-Aerospace from the von KarmanInstitute for Fluid Dynamics, Belgium. He received his PhD in 1985 from the Faculty ofMechanical Engineering, RWTH Aachen University, Germany. During 1985–1987, he wasa postdoc at FFA, The Aeronautical Research Institute of Sweden. He served as a ResearchScientist at the Institute for Theoretical Fluid Mechanics of DLR in Göttingen 1987–1990.During 1990–1996, he was appointed as an Assistant Professor at the Institute of FluidDynamics of ETH Zurich, Switzerland, where he did his habilitation in theoretical fluidmechanics in 1997. After stays at the Faculty of Applied Mathematics, University of Twente,The Netherlands, 1996–1997, and at the Department of Mathematics of Luleå Universityof Technology, Sweden, 1997–1998, he joined Uppsala University, Sweden, as an AssociateProfessor at the Department of Information Technology 1998–2007. Since 2007, he hasbeen a Professor of Computational Fluid Dynamics (CFD) at the Department of Energyand Process Engineering of the Norwegian University of Science and Technology (NTNU).His research interests include CFD, computational aeroacoustics, fluid-structure interaction,bio-fluid dynamics and multiphase flow simulation.

Copyright © 2012 Inderscience Enterprises Ltd.

High-order numerical simulation of fluid-structure interaction in the human larynx 165

1 Introduction

Our voice is generated in our larynx by the vibratingvocal folds interacting with the airstream from the lungs.The vocal folds, or vocal cords, are two symmetricmembranes that protrude from the walls of the larynxat the top of the trachea of humans and most mammalsforming a slit-like opening known as the glottis inthe airway. In a simplified three-layer model, the vocalfolds are composed of the thyroarytenoid muscle, alsoknown as the vocal fold muscle, and the vocal ligamentcovered by a mucous layer, cf. Figure 1. During normalbreathing, the vocal tract is open and air can passunobstructedly. During phonation, the vocal fold muscleis tensed in the longitudinal direction so that the glottalopening becomes narrower. As the high-pressure airexpelled from the lungs is forced through this narrowopening, it pushes the vocal folds apart. The air columngains momentum, and the velocity in the glottis increases.The increase in velocity causes a pressure drop in theglottis according to the Bernoulli principle. The decreasein pressure leads to an aerodynamic force, which togetherwith the elastic force in the vocal folds strives to closethe glottis. A build-up in pressure upstream of the glottisresulting from the closure leads to a pressure force,which opens the vocal folds and allows the passageof air. Under favourable circumstances, this process isrepeated in a self-sustained manner and driven only bythe pressure from the lungs. It is important to point outthat no periodic contraction of muscles occurs duringphonation. The opening and closing of the glottis is inthis respect a passive process (Titze, 2000).

Figure 1 Schematic view of the human larynx (see onlineversion for colours)

During normal speech, the vocal folds collide with eachother, closing the glottis completely. However, in certaintypes of phonation such as a very breathy voice or whilewhispering or hissing, the vocal folds do not necessarilymake contact. The outermost mucous layer of the vocalfolds have been shown to play an important role in theself-sustained oscillation, facilitating the vibrations of themuch stiffer ligament (Titze, 2000).

As the vocal folds oscillate rapidly, they generatea fundamental frequency. When we speak normally,only the lowest mode of vibration is excited, in whichall the layers of the vocal folds vibrate symmetricallyand as a whole. Higher modes of oscillation can,

however, be excited to produce higher pitched tonesfor example when singing. By stretching the vocal foldmuscle, the vocal fold length changes as well as thestiffness, and higher modes can thus be created. Thesehigher modes correspond to an oscillation concentratedmainly to the ligament or the mucous layer and have ahigher frequency. The different modes of oscillation arecommonly referred to as registers (modal, falsetto, etc.)and singers are often particularly good at smoothing outthe transition between these registers (Titze, 2008).

The acoustic signal resulting from the glottal flowinteracting with the vibrating vocal folds is furthermodified by the vocal tract, which functions as anacoustic filter. By changing the shape of the vocal tract,different frequencies are amplified and suppressed so thata multitude of different vowels can be formed from thesame source signal.

The airflow and vocal folds in the larynx areintrinsically 3D and would need to be modelled assuch to obtain a reliable flow field downstream of theglottis. However, the shapes exhibited by the glottis atdifferent time instants during the vibratory cycle showlittle variation in the transverse direction of the vocalfold. Especially in configurations when the glottis isalmost closed, the slit-like opening formed by the twovocal folds can be considered as two-dimensional. Inconfigurations where the glottis is fully open, transverseeffects may be important and a 2D model might not beable to accurately represent the flow field during thoseparts of the cycle. However, since most of the vortexdynamics takes place during the almost closed phases,the acoustic pressure variations emitted by those vorticescan be approximately represented in a 2D geometry.

The computational models for self-sustainedvibrations of vocal folds have advanced from simplemodels based on the Bernoulli equation for the airflowand mass-spring models for the vocal folds used inthe 1970s to full-fledged 2D and 3D unsteady Fluid-Structure Interactions (FSIs) coupling the Navier–Stokesequations and Navier’s equation (Titze, 2000; Luo et al.,2008; Link et al., 2009). In general, lower-order finitedifference, volume and element methods have been used.

In this paper, we employ a high-order finite differenceapproach based on SBP operators (Strand, 1994;Gustafsson et al., 1995; Gustafsson, 2008) to solve thecompressible Navier–Stokes equations and the linearelastic wave equation, i.e., Navier’s equation. Fluid andstructure interact in a two-way coupling, meaning thatfluid stresses deform the flexible structure, which inturn causes the fluid to conform to the new structuralboundary via no-slip boundary conditions. The tractionboundary conditions and the location and velocity ofthe vocal fold boundaries are communicated betweenstructure and fluid at the beginning of each time stepof the explicit Runge-Kutta time stepping of fluid andstructure. While the velocity and traction boundaryconditions for the structure are weakly imposed usingthe SAT approach (Larsson and Müller, 2011), theno-slip boundary conditions for the fluid are enforced

166 M. Larsson and B. Müller

by injecting the data supplied by the structure. Theapproach has been tested for a 2D model of the larynxand the vocal folds.

Like the other methods for FSI in the human larynxcited earlier, the present method cannot handle contact ofthe vocal folds occurring during their collision, becauseit cannot handle cells with zero volume. We thereforeconsider a case where the vocal folds do not collide. Asmentioned earlier, such situations occur in nature forbreathy voices and when whispering or hissing soundsare produced. To model typical human phonation, acontact model and a special treatment for the closingglottis would be needed.

The present work is based on Martin Larsson’s PhDthesis (Larsson, 2010) and the six papers by Larssonand Müller (2009a, 2009b, 2011, 2010a, 2010b) containedtherein.

2 Governing equations

2.1 Compressible Navier–Stokes equations

The perturbation formulation is used to minimisecancellation errors when discretising the Navier–Stokesequations for compressible low Mach number flow(Sesterhenn et al., 1999; Müller, 1996). The 2Dcompressible Navier–Stokes equations in conservativeform can be expressed in perturbation form as Müller(2008) and Larsson and Müller (2009a)

U ′t + F c′

x + Gc′y = F v ′

x + Gv ′y, (1)

where the vector U ′ denotes the perturbation of theconservative variables with respect to the stagnationvalues. U ′ and the inviscid (superscript c) and viscous(superscript v) flux vectors can be found in, e.g., Larssonand Müller (2009a).

General moving geometries are treated by atime-dependent coordinate transformation τ = t, ξ =ξ(t, x, y), η = η(t, x, y). The transformed 2D conservativecompressible Navier–Stokes equations in perturbationform read (Larsson and Müller, 2009a)

U ′τ + F ′

ξ + G′η = 0, (2)

where U ′ = J−1U ′, F ′ = J−1(ξtU′ + ξx(F c′ − F v ′) +

ξy(Gc′ − Gv ′)) and G′ = J−1(ηtU′ + ηx(F c′ − F v ′) +

ηy(Gc′ − Gv ′)). Equation (2) constitutes the ALEformulation of the 2D compressible Navier–Stokesequations in perturbation form.

No-slip adiabatic wall boundary conditions are usedat the upper and lower walls of the vocal tract includingthe moving boundaries of the upper and lower vocalfolds, cf. Figure 2. The Navier–Stokes CharacteristicBoundary Conditions (NSCBC) technique by Poinsotand Lele (1992) is employed at the outflow, i.e., theright boundary in Figure 2 (Larsson and Müller, 2009b).At the inflow, i.e., the left boundary in Figure 2, thepressure, temperature and y-component of velocity are

imposed as p = patm + ∆p, T = T0 = 310 K, and v = 0,respectively. The x-component of velocity u at the inflowand the pressure p at the walls are computed from the2D compressible Navier–Stokes Equations (2) discretisedat the boundaries.

Figure 2 Computational fluid and structure domains ofairflow and vocal folds, respectively (see onlineversion for colours)

2.2 Linear elastic wave equation

The 2D linear elastic wave equation written as a first-order hyperbolic system reads in Cartesian coordinates

qt = Aqx + Bqy, (3)

where the unknown vector q = (u, v, f, g, h)T containsthe velocity components u, v and the stress componentsf, g, h. The coefficient matrices A and B depend on thethe Lamé parameters λ, µ and the density ρ, which arehere all taken to be constant in space and time, cf. e.g.,Fornberg (1998) and LeVeque (2002).

The linear combination P (kx, ky) = kxA + kyB canbe diagonalised with real eigenvalues and linearlyindependent eigenvectors. The eigenvalue matrix isdefined as the diagonal matrix with the eigenvalues ofP (kx, ky) in decreasing order,

Λ(kx, ky) = (k2x + k2

y)1/2diag(cp, cs, 0,−cs,−cp)

= diag

λi(kx, ky)5

i=1, (4)

where the wave speeds are cp =√

(λ + 2µ)/ρ and cs =√µ/ρ, referred to as primary (or pressure) and secondary

(or shear) wave speeds, respectively.To treat curvilinear grids, we introduce the mapping

x = x(ξ, η), y = y(ξ, η). The Jacobian determinant J−1

of the transformation is given by J−1 = xξyη − xηyξ andthe linear elastic wave equation can then be written as

qt = (Aq)ξ + (Bq)η (5)

where the hats signify that the quantities are intransformed coordinates, i.e., q = J−1q, A = ξxA + ξyB

and B = ηxA + ηyB.


2.3 Characteristic variables

To describe the SAT expressions in transformedcoordinates we need to find the characteristic variablesfor the transformed equation in which the coefficientmatrices are linear combinations of the coefficientmatrices in the x- and y-directions.

qt = ((kxA + kyB)q)k (6)

for k = ξ, η. We form the linear combination P (kx, ky) =kxA + kyB. The coefficient matrices A and B havethe same set of eigenvalues Λ = diag(cp, cs, 0,−cs,−cp),whereas for the linear combination P (kx, ky) weget Λ(kx, ky) = (k2

x + k2y)1/2Λ. To find the linearly

independent eigenvectors of P (kx, ky), we solve theunder-determined system (P (kx, ky) − λiI)vi = 0 for i =1, . . . , 5. These five eigenvectors vi become the columnsin the eigenvector matrix T (kx, ky), cf. Appendix A.We have some degrees of freedom in choosing T ,because each column can be scaled by any non-zeroconstant. The inverse of this matrix is obtained with asymbolic computer program, cf. Appendix A where wehave introduced the following abbreviations k = (k2

x −k2

y)/(k2x + k2

y), r = (k2x + k2

y)1/2, cp = rcp, cs = rcs, α =(λ + 2µ)/λ and β = αλ/µ. For all directions (kx, ky)we have that T−1(kx, ky)P (kx, ky)T (kx, ky) = Λ(kx, ky).The transformation to characteristic variables u is givenby u(k) = T−1(kx, ky)q for each of the two coordinatedirections k = ξ, η. The transformation back to flowvariables is given by q = T (kx, ky)u(k).

3 Time stable high order difference method

3.1 Energy method

The energy method is a general technique to provesufficient conditions for well posedness of PartialDifferential Equations (PDEs) and stability of differencemethods with general boundary conditions.

Consider the solution of the model problem in 1Dwith

ut = λux, λ > 0, 0 ≤ x ≤ 1, t ≥ 0,

u(x, 0) = f(x), u(1, t) = g(t). (7)

Here, the symbol λ represents a general eigenvalue forthe hyperbolic system and should not be confused withthe Lamé parameter. Define the L2 scalar product forreal functions v and w on the interval 0 ≤ x ≤ 1 as

(v, w) =∫ 1

0v(x)w(x)dx (8)

which defines a norm of the continuous solution atsome time t and an energy E(t) = ‖u(·, t)‖2 = (u, u).Using integration by parts (v, wx) = v(1, t)w(1, t) −v(0, t)w(0, t) − (vx, w), we get dE

dt = d‖u‖2

dt = (ut, u)+(u, ut) = λ[(ux, u) + (u, ux)] = λ[(ux, u) + [u2]10 − (ux,

u)] = λ[u2(1, t) − u2(0, t)]. If λ > 0, the boundarycondition u(1, t) = 0 yields a non-growing solution (notethat periodic boundary conditions would also yielda non-growing solution), i.e., E(t) ≤ E(0) = ‖f(x)‖2.Thus, the energy of the solution is bounded by theenergy of the initial data. As a unique solution ofthe Initial-Boundary Value Problem (IBVP) (7) exists,the problem is well posed.

3.2 Summation by parts operators

The idea behind the SBP technique for first-order IBVP isto devise difference approximations Q of the first spatialderivative satisfying the discrete analogue of integrationby parts called the SBP property (Gustafsson, 2008). Tooutline the idea for the numerical solution of (7), weintroduce the equidistant grid xj = jh, j = 0, . . . , N , h =1/N , and a solution vector containing the solution at thediscrete grid points, u = (u0(t), u1(t), . . . , uN (t))T . Thesemi-discrete problem can be stated using a differenceoperator Q approximating the first derivative in space,

dudt

= λQu, uj(0) = f(xj). (9)

We also define a discrete scalar product andcorresponding norm and energy by

(u,v)h = h∑i,j

hijuivj = huTHv,

Eh(t) = ‖u‖2h = (u,u)h, (10)

where the symmetric and positive definite norm matrixH = diag(HL, I, HR) has components hij . In order forEquation (10) to define a scalar product, HL and HR

must be symmetric and positive definite. We say that thedifference operator Q satisfies the SBP property, if

(u, Qv)h = uNvN − u0v0 − (Qu,v)h. (11)

It can be seen that this property is satisfied, if thematrix G = HQ satisfies the condition that G + GT =diag(−1, 0, . . . , 0, 1). If Q satisfies the SBP property (11),then the energy method for the discrete problem yields:

dEh

dt=

d‖u‖2h

dt= (ut,u)h + (u,ut)h

= λ[(Qu,u)h + (u, Qu)h]= λ[(Qu,u)h + u2

N − u20 − (Qu,u)h]

= λ[u2N − u2

0]. (12)

How to obtain time stability dEh/dt ≤ 0, i.e., no energygrowth in time, is the topic of the next section.

For diagonal HL and HR, there exist differenceoperators Q accurate to order O(h2s) in the interior andO(hs) near the boundaries for s = 1, 2, 3 and 4. Theseoperators have an effective order of accuracy O(hs+1)in the entire domain. Explicit forms of such operators Qand norm matrices H were derived by Strand (1994).

For this study, we use an SBP operator based onthe central sixth-order explicit finite difference operator


(s = 3), which has been modified near the boundaries tosatisfy the SBP property giving an effective O(h4) orderof accuracy in the whole domain (Strand, 1994).

3.3 Simultaneous approximation term

Since the term λu2N in Equation (12) is non-negative,

time stability does not follow when using the injectionmethod for the SBP operator, i.e., by using uN (t) =g(t), since injection affects the operator Q and theSBP property (11) (Strand, 1994; Gustafsson, 2008). Incontrast, the SAT method by Carpenter et al. (1994) isan approach where a linear combination of the boundarycondition and the differential equation is solved atthe boundary. This leads to a weak imposition of thephysical boundary condition. The imposition of SATboundary conditions is accomplished by adding a sourceterm to the difference operator, proportional to thedifference between the value of the discrete solution uN

and the boundary condition. The SAT method for thesemidiscrete advection Equation (9) can be expressed as

dudt

= λQu − λτS(uN − g(t))

where S = h−1H−1(0, 0, . . . , 0, 1)T and τ is a freeparameter.

The added term does not alter the accuracy ofthe scheme, since it vanishes when the analyticalsolution is substituted. Thus, we can imagine the SATexpression as a modification to the difference operatorso that we are effectively solving an equation ut = λQuwith Q = Q + Qsat where the boundary conditionsare accounted for by the operator itself. When H isdiagonal, the scheme is only modified at one point,namely at the point where the boundary condition isimposed. We can now show that this scheme is timestable for g(t) = 0. The energy rate for the solution of

the semi-discrete equation is dEh

dt = d‖u‖2h

dt = (ut,u)h +(u,ut)h = λ[(Qu − τSuN ,u)h + (u, Qu − τSuN )h] = λ[(Qu,u)h − τ(S,u)huN + (u, Qu)h − τ(u,S)huN ] = λ[(1 − 2τ)u2

N − u20] since (S,u)h = (u,S)h = huTHh−1

H−1(0, 0, . . . , 0, 1)T = uN . The discretisation is timestable if τ ≥ 1/2.

The extension of the time stable SAT method to 1Dhyperbolic systems

ut = Λux (13)

with a diagonal r × r coefficient matrix Λ is performedin the following way (Carpenter et al., 1994). Thecoefficient matrix Λ is chosen such that the diagonalentries appear in descending order, i.e., λ1 > λ2 >· · · > λk > 0 > λk+1 > · · · > λr. The solution vector uis split into two parts corresponding to positive andnegative diagonal elements uI = (u(1), . . . , u(k))T anduII = (u(k+1), . . . , u(r))T , where u(i) is the ith componentof u. Since the ith component of (13) reads u

(i)t = λiu

(i)x ,

the vectors uI and uII are transported to the left andright, respectively. Therefore, boundary conditions have

to be prescribed on uI at the right boundary x = 1 andon uII at the left boundary x = 0. To allow for couplingof the in- and outgoing variables at the boundaries,we introduce a k × (r − k) matrix R and a (r − k) ×k matrix L. We define boundary functions gI(t) =(g(1)(t), . . . , g(k)(t)) and gII(t) = (g(k+1)(t), . . . , g(r)(t)).Then, the boundary conditions are given by

uI(1, t) = RuII(1, t) + gI(t),uII(0, t) = LuI(0, t) + gII(t). (14)

Under the constraint |R‖L| ≤ 1, the IBVP (13) with (14)is well posed (Carpenter et al., 1994), where the matrix2-norm is defined by |R| =

√ρ(RT R) and ρ(A) is the

spectral radius of A.We define the grid functions of the components

of u as u(i) = (u(i)0 , . . . , u

(i)N )T , where u

(i)j = u(i)(xj).

Then, we define the grid functions of uI and uII asuI = (u(1), . . . ,u(k)) and uII = (u(k+1), . . . ,u(r)). Theboundary conditions (14) for the semi-discretisation of(13) are imposed by the SAT method as Carpenter et al.(1994)

du(i)

dt = λiQu(i) − λiτS(i)(u(i)N −

(RuIIN )(i) − g(i)(t)), 1 ≤ i ≤ k

du(i)

dt = λiQu(i) + λiτS(i)(u(i)0 −

(LuI0)

(i−k) − g(i)(t)), k + 1 ≤ i ≤ r

(15)

where S(i) = h−1H−1(0, 0, . . . , 1)T for 1 ≤ i ≤ kand S(i) = h−1H−1(1, 0, . . . , 0)T for k + 1 ≤ i ≤ r.Regarding the notation, (RuII

N )(i) should be interpretedas follows: uII

N = (u(r−k)N , . . . , u

(r)N )T is the last row of uII

transposed. Multiplying R by uIIN yields a new vector of

which the (i)th component is taken. The interpretationof uI

0 is similar with uI0 = (u(1)

0 , . . . , u(k)0 )T . As shown by

Carpenter et al. (1994), the SAT method is both stablein the classical sense and time stable provided that

1 −√

1 − |R‖L||R‖L| ≤ τ ≤ 1 +

√1 − |R‖L|

|R‖L| . (16)

4 Sat expressions for the linear elastic wave equation

Notation for boundary conditions

We adopt the notation u(k0, t) = u(k = k0, t) torepresent a 1D boundary condition on the solutionvariable u in any direction k where k = ξ or k = η andu(k, t) is the given functions of time on the boundariesk0 = 0 and k0 = 1, which the solution variable u shouldmatch on those boundaries. For example, u(ξ = 1, t) isthe given u-velocity at the boundary ξ = 1 and u(1, t)is the corresponding solution to the equations. In 2D,the boundary condition also depends on the secondcoordinate direction, which we indicate by u(ξ = 1, η, t)and u(ξ, η = 1, t) for boundary conditions in the ξ- and


η-directions, respectively. Finally, for the discretised 2Dboundary conditions, we write instead uj(ξ = 1, t) =u(ξ = 1, ηj , t) and ui(η = 1, t) = u(ξi, η = 1, t).

4.1 Presentation of SAT expressions

The SAT expressions for the linear elastic wave equationare summarised here.

To apply the theory for 1D linear diagonalisedhyperbolic systems (13), we have to express (6)in characteristic form, i.e., ut = Λuk. u = u(k) =T−1(kx, ky)q is the vector of the characteristic variablesin the k-direction where q = J−1(u, v, f, g, h)T . Λ =Λ(kx, ky) is the diagonal matrix with the eigenvalues ofP (kx, ky) where k = ξ or k = η, and uk = ∂u

∂k .We form the two sub-vectors corresponding to

positive and negative eigenvalues as

uI(kx, ky) = (u1, u2)T , uII(kx, ky) = (u4, u5)T (17)

with the aim to form boundary conditions with thematrices R and L. Since the components of uI anduII contain velocity and stress components in positiveor negative pairs, it is easy to find 2 × 2 matrices Land R in Equation (14) to prescribe velocity or tractioncomponents as boundary conditions gI and gII. Letus illustrate this approach for prescribing the velocitycomponents u and v at the boundaries.

Looking at the characteristic variables, we find thefollowing identities to prescribe the velocity componentsun = (kxu + kyv)/r and ut = (−kyu + kxv)/r normaland tangential, respectively, to a grid line k = const:

uI(kx, ky) −[

0 1−1 0

]uII(kx, ky) =

J−1

r

[λcp

un

ut

](18)

uII(kx, ky) −[

0 −11 0

]uI(kx, ky) =

J−1

r

[ut

− λcp

un

](19)

Thus, using the matrices L =[0 −11 0

]and R =

[0 1

−1 0

]in

Equation (14), we can prescribe boundary conditions for

gI(kx, ky, t) = J−1

r2

[ λcp

(kxu + kyv)−kyu + kxv

]and gII(kx, ky, t) =

J−1

r2

[ −kyu + kxv− λ

cp(kxu + kyv)

], cf. (20) and (21) below. Note

that L and R are independent of the direction, butdepend on the particular type of boundary conditionto be imposed (velocity or traction). For boundaryconditions on the velocities u and v, we get using theidentities (18)–(19) the following expressions

R =

[0 1

−1 0

], gI(kx, ky, t)

=J−1

r2

[λcp

(kxu(k = 1, t) + ky v(k = 1, t))

−kyu(k = 1, t) + kxv(k = 1, t)

], (20)

L =[

0 −11 0

], gII(kx, ky, t)

=J−1

r2

[ −kyu(k = 0, t) + kxv(k = 0, t)

− λcp

(kxu(k = 0, t) + ky v(k = 0, t))

], (21)

where u(k, t), v(k, t) are the given boundary conditionson u, v at the boundaries.

The boundary conditions on the stresses come froma traction boundary condition of the form σn = twhere t = (tx, ty)T is the given traction vector from the

fluid and σ =[f gg h

]is the Cauchy stress tensor in the

structure. The unit normal n can be expressed in terms ofthe coordinate transformation as n = (1/r)(kx, ky)T , cf.Figure 3, and the components of gI and gII for tractionboundary conditions can be written as

R =[

0 −11 0

], gI(kx, ky, t) =

J−1

r2

×[ 1

αr (kxtx(k = 1, t) + ky ty(k = 1, t))r

ρcs(−ky tx(k = 1, t) + kxty(k = 1, t))

], (22)

L =[

0 1−1 0

], gII(kx, ky, t) =

J−1

r2

×[

rρcs

(ky tx(k = 0, t) − kxty(k = 0, t))1

αr (kxtx(k = 0, t) + ky ty(k = 0, t))

]. (23)

Therefore, it is sufficient to specify the two parameterstx and ty on each boundary instead of all of the threef , g, h, which might otherwise violate well posedness.

Figure 3 Coordinate transformation

Inserting the definitions of gI,II and R, L gives withEquation (15) a SAT expression (which we simply callSAT) for each of the five equations in characteristicvariables. At a general index i ∈ 0, 1, . . . , N in thek-direction, the SAT vector for prescribed velocitycomponents will be SAT

(k)i (kx, ky, t) = − τJ−1

hkr2

λh−1NNδiN [kx(uN − u(k = 1, t)) + ky(vN − v(k = 1, t))]

cskh−1NNδiN [−ky(uN − u(k = 1, t)) + kx(vN − v(k = 1, t))]

0cskh−1

00 δi0[−ky(u0 − u(k = 0, t)) + kx(v0 − v(k = 0, t))]−λh−1

00 δi0[kx(u0 − u(k = 0, t)) + ky(v0 − v(k = 0, t))

,

where h00 and hNN are the first and last entries,respectively, of the diagonal norm matrix H . Note thatthe SAT term for the characteristic variable u3 with


characteristic speed zero is zero, because for u3 noboundary condition must be given. For k = ξ, we use theindex i, while for k = η we employ index j for i and Mfor N . Equation (16) implies τ = 1.

For each of the two spatial directions, thetransformation matrix T (kx, ky) is applied to get thecorresponding SAT expressions in flow variables.

SAT(k)i = T (kx, ky)SAT(k)

i (kx, ky) (24)

for k = ξ and η. Finally, the total SAT expression isthen the sum of the two contributions from the twocoordinate directions.

SATi,j = SAT(ξ)i,j (ξx, ξy) + SAT

(η)i,j (ηx, ηy). (25)

5 Fluid-structure interaction

5.1 Arbitrary Lagrangean–Eulerian (ALE)formulation

The displacement of the fluid-structure interfacedetermines the shape of the fluid domain, and thestructure velocity at the interface determines the internalgrid point velocities in the fluid domain. The left andright boundaries of the fluid domain are the in- andoutflow, respectively. The top and bottom parts of thefluid domain are bounded by the flexible vocal folds andthe inner wall of the vocal tract, which is here assumedto be rigid. As we do not assume symmetry with respectto the streamwise centreline of the vocal tract, the twovocal folds are solved for individually. In our ALEformulation, the positions and velocities of the gridpoints in the fluid domain are linearly interpolated alongthe grid lines connecting the upper and lower vocalfolds where the positions and velocities are given by thestructure solution. Figure 4 shows the given structurevelocities with bold arrows and the interpolated gridpoint velocities x, y (thin arrows) for three grid lines.To obtain the time derivative of J−1 as needed in (2), ageometric invariant (Visbal and Gaitonde, 2002) is used.This geometric conservation law states that

(J−1)τ + (J−1ξt)ξ + (J−1ηt)η = 0. (26)

Figure 4 The boundary of the fluid domain consists of fixedand flexible parts. The velocity at the boundary ofthe flexible part determines the internal grid pointvelocity. Only the lower half of the domain isshown

The time derivatives of the computational coordinatesξ, η can here be obtained from the grid point velocitiesx, y as ξt = −(xξx + yξy), ηt = −(xηx + yηy), which canbe seen by differentiating the transformation with respectto τ . With the ξ- and η-derivatives in Equation (26)discretised by the globally fourth-order SBP operator,we get the time derivative (J−1)τ at each timelevel. The Jacobian determinant J−1 of the coordinatetransformation is determined by J−1 = xξyη − xηyξ andthe metric terms by J−1ξx = yη, J−1ξy = −xη, J−1ηx =−yξ, J−1ηy = xξ.

5.2 Description of fluid-structureinteraction algorithm

At the start of a simulation, we construct the fixedreference configuration for the structure and set theinitial variables to zero (zero velocity and no internalstresses). The initial conditions for the perturbationvariables U ′ in the fluid domain are taken equal to zeroas well (stagnation conditions). In the first time step,the fluid domain is uniquely determined by the referenceboundary of the structure. To go from time level n ton + 1, we first take one time step for the fluid withimposed pressure boundary conditions at the inflow andadiabatic no-slip conditions on the walls, i.e., u = uw and∂T/∂n = 0. After the fluid time step, the fluid stress onthe wall is calculated based on the new fluid velocitiesand pressures. These fluid stresses σf are passed on tothe structure solver via the traction boundary condition.The force per unit area exerted on a surface element withunit normal n is t = σfn, where n is here the outer unitnormal of the structure, calculated from the displacementvector field.

The traction computed at time level n for the fluidis then used to advance the structure solution to timelevel n + 1. Note that the traction tn is used, although(σf )n+1nn ≈ tn+1 is available. For we employ explicittime integration where we start from time level n forboth structure and fluid. The solution for the structure atthe new time level gives the velocities and displacementson the boundary, which in turn are used to generate thenew fluid grid and internal grid point velocities. Thisprocedure is repeated for each time step.

The fluid-structure interaction algorithm issummarised as follows:

1 Generate the initial fluid grid based on the referenceconfiguration for the structure. ⇒ x0, x0.

2 Give initial values for the fluid and the structure.⇒ F 0, S0.

3 For time step n = 1, 2, . . . , do:

(a) Calculate the fluid stress on the boundary andcalculate the force per unit area, i.e., traction,on the structure via the unit normal. Store thetraction vector tn, cf. Figure 5(a).


Figure 5 Illustration of fluid-structure interaction algorithm:Ωf , Ωs: fluid and structure domains. (a) Thetraction vector T(x, t) = t(x, t) exerted on thestructure by the fluid is calculated on thefluid-structure interface as the fluid stress tensortimes the outward unit normal. (b) The grid pointvelocity xi,j at grid point (i, j) is interpolated fromthe given velocity v of the structure on the interface

(b) Take one time step for the fluid:Fn+1 = F (xn, xn).

(c) Then the new fluid pressure pn+1 and the newfluid velocity un+1 on the boundary areavailable.

(d) Take one time step for the structure using theboundary conditions tn: Sn+1 = S(tn).

(e) Recalculate the fluid grid and the grid pointvelocities based on the new structure solution,cf. Figure 5(b). ⇒ xn+1, xn+1.

4 Repeat from 3 with time step n + 1 until the finaltime is reached.

6 Discretisation

Notation

The Kronecker product of an n × m matrix C and ak × l matrix D is the n × m block matrix

C ⊗ D =

c11D · · · c1mD...

. . ....

cn1D · · · cnmD

. (27)

This notation will be useful for writing the discretisationin a compact form.

6.1 Linear elastic wave equation

Introduce a vector q = (qijk)T = (q001, . . . , q005, q101, . . . , q105, . . . , qNM5)T where the three indices i, j and krepresent the ξ-coordinate, η-coordinate and the solutionvariable, respectively. We define difference operators interms of Kronecker products that operate on one indexat a time.

Let Qξ = Qξ ⊗ IM ⊗ I5 and Qη = IN ⊗ Qη ⊗ I5where Qξ and Qη are 1D difference operators inthe ξ- and η-directions, respectively, satisfying theSBP property (11). The identity operators IN andIM are unit matrices of size (N + 1) × (N + 1) and(M + 1) × (M + 1), respectively. The computation ofthe spatial differences of q can then be seen as operatingon q with one of the Kronecker products, i.e., Qηqoperates on the second index and yields a vector ofthe same size as q representing the first derivativeapproximation in the η-direction. To express the semi-discrete linear elastic wave equation, we also needto define A = IN ⊗ IM ⊗ A and B = IN ⊗ IM ⊗ B.Note that these products are never actually explicitlyformed as they are merely theoretical constructs to makethe notation more compact. The products correspondwell to the actual finite difference implementation,i.e., the approximations of the first derivatives arecalculated by operating on successive lines of valuesin the computational domain. Using the Kroneckerproducts defined earlier, the semi-discrete linear elasticwave equation with constant coefficients including theSAT expression can be written as

dqdt

= Qξ(Aq) + Qη(Bq) + SAT (28)

where SAT is the SAT expression in transformedcoordinates defined in Equation (25).

6.2 Navier–Stokes equations

For the fluid equations, we employ a similar procedure,i.e., we define vectors for the solution variables U′ =(U ′

ijk)T = (U ′001, . . . , U

′004, U

′101, . . . , U

′104, . . . , U

′NM4)

T,

and similarly for the two flux vectors F′ and G′,where again the three indices i, j and k represent theξ-coordinate, η-coordinate and the solution variable,


respectively. The same difference operators are used asfor the linear elastic wave equation. The discretised fluidequation can thus be written as

dU′

dτ= −QξF′ − QηG′. (29)

6.3 Time integration

The systems (28) and (29) of ordinary differentialequations can readily be solved by the classical fourth-order explicit Runge–Kutta method. For the linearelastic wave equation, calling the right-hand side of (28)f(tn, qn) at the time level n, we advance the solution tolevel n + 1 by performing the stages

k1 = f(tn, qn)

k2 = f(

tn +∆t

2, qn +

∆t

2k1

)k3 = f

(tn +

∆t

2, qn +

∆t

2k2

)k4 = f(tn + ∆t, qn + ∆tk3)

qn+1 = qn +∆t

6(k1 + 2k2 + 2k3 + k4)

and similar expressions for the fluid Equation (29). Theboundary conditions are updated only after all fourstages for the respective field have been completed. Thatis to say, the structure solution at level n + 1 is obtainedusing only the fluid stress at time level n. Likewise, thefluid solution at time level n + 1 is based only on theposition and velocity of the structure at time level n.

7 Results

Verification

Our fluid solver has previously been verified and testedfor numerical simulation of Aeolian tones (Müller,2008) and qualitatively tested for simulation of humanphonation on fixed grids (Larsson and Müller, 2009a)as well as moving grids in ALE formulation (Larsson,2007).

The solver for the linear elastic equations with theSAT expression has been tested with a manufacturedsolution (Larsson and Müller, 2011) and an academic2D test case where we obtained a rate of convergence of3.5–4 in the 2-norm (to be published).

7.1 Problem parameters

The initial geometry for the vocal folds is here based onthe geometry used in Zhao et al. (2002) for an oscillatingglottis with a given time dependence. The geometrychosen by Zhao et al. (2002) is an idealised vocal tractwith a time-varying geometry, where the false vocal foldsare not taken into account. The initial shape of the vocaltract including the vocal fold is given as

rw(x) =D0 − Dmin

4tanh s +

D0 + Dmin

4, (30)

where rw is the half height of the vocal tract, D0 =5Dg is the height of the channel, Dg = 4mm is theaverage glottis height, Dmin = 1.6mm is the minimumglottis height, s = b|x|/Dg − bDg/|x|, c = 0.42 and b =1.4. For −2Dg ≤ x ≤ 2Dg , the function (30) describesthe curved parts of the reference configuration for thetop and bottom (with a minus sign) vocal folds. Thex-coordinates for the in- and outflow boundaries are−4Dg and 10Dg , respectively.

7.2 Vocal fold material parameters

The density in the reference configuration is ρ0 =1043 kg/m3, corresponding to the measured density ofvocal fold tissue as reported by Hunter et al. (2004).The Poisson ratio is chosen as ν = 0.47 for the tissue,corresponding to a nearly incompressible material withν = 0.5 being the theoretical incompressible limit. TheLamé parameters are chosen as µ = 3.5 kPa and λ givenby λ = 2µν/(1 − 2ν).

7.3 Fluid model

We use a Reynolds number of 3000 based on theaverage glottis height Dg = 0.004 m and an assumedaverage velocity in the glottis of Um = 40 m/s. Weemploy these particular values to be able to comparewith previously published results by Zhao et al. (2002)and Zhang et al. (2002) and by (Larsson, 2007) andLarsson and Müller (2009a). The Prandtl number isset to 1.0, and the Mach number is 0.2 based onthe assumed average velocity and the speed of sound.We deliberately use a lower value for the speed ofsound c0 = 200 m/s to speed up the computations. Weimplemented the higher Mach number by using thestagnation density ρ0, the lowered stagnation speed ofsound c0 and ρ0c

20 as reference values of the non-

dimensional density, velocity and pressure, respectively.The air density is 1.3 kg/m3, and the atmosphericpressure is patm = 101325 Pa. The equation of state isthe perfect gas law, and we assume a Newtonian fluid.At the inlet, we impose a typical lung pressure duringphonation with a small asymmetric perturbation bysetting the acoustic pressure to pacoustic = p − patm =(1 + 0.025 sin 2πη)2736 Pa, where η = 0 at the lowervertex and η = 1 at the upper vertex of the inflowboundary. The outlet pressure is set to atmosphericpressure, i.e., p − patm = 0 Pa.

7.4 Numerical simulation

Both fluid and structure use the same set of variables fornon-dimensionalisation, and the same time step is usedfor both fields so that the two solutions can exchangeinformation at the same time levels. The structure gridconsists of 81 × 61 points for each vocal fold, i.e., forthe upper and the lower vocal folds, and the fluid gridhas 241 × 61 points. The time step is determined by thestability condition for the fluid, which is satisfied here by


requiring CFL ≤ 1. Since the fluid domain changes withtime, the CFL condition puts a stricter constraint on thetime step when the glottis is nearly closed. The solutionis marched in time with given initial and boundaryconditions to dimensional time t = 12 ms (total numberof time steps 277310).

The acoustic pressure is recorded at four pointsat x = 8 Dg . Points 1–4 are located at y = −3D0/10,y = −D0/10, y = D0/10, and y = 3D0/10, respectively.Figure 6 shows the computed non-dimensional acousticpressure p′/(ρ0c

20) at the four points as a function of

the non-dimensional time t = tc0/Dg for a durationof 7ms in physical time. Large acoustic pressures arerecorded as the start-up vortex passes the points duringthe time interval of about 40 ≤ t ≤ 90. The modulusof the acoustic pressure decreases as large vortices passat t ≈ 150 and t ≈ 240. Smaller vortices cause smallervariations of the acoustic pressure. Since the acousticpressure curves recorded at the four points lie almoston top of each other, the acoustic pressure at x = 8 Dg

slightly upstream of the right boundary proves to behavealmost one-dimensionally.

Figure 6 Non-dimensional acoustic pressure p′/(ρ0c20) as a

function of non-dimensional time t = tc0/Dg at 4points located at x = 8 Dg and y = −3D0/10,y = −D0/10, y = D0/10, and y = 3D0/10,respectively

The solution is first integrated to time t = 6 ms so thatthe effect of initial conditions will be negligible. Afterthat, the solution is recorded at consecutive 2 ms intervalsas shown in Figure 7 where the vorticity and pressurecontours are depicted in the left and right columns,respectively.

Initially, a starting jet is formed in the glottis,which becomes unstable near the exit and creates thebeginnings of vortical structures at time t = 6 ms. Sincethe boundary conditions are not symmetric with respectto the streamwise centreline, also the solution is notsymmetric. In the following, vortices are shed near theglottis and propagate downstream driven by the pressuregradient. The pressure plots indicate a sharp pressuredrop just upstream of the orifice. Downstream, thepressure minima occur in the vortex centres as expected.

The observed frequency of the vortex sheddingis about 80 Hz, which is comparable to the typicalphonation frequencies of 100 Hz for men and 200 Hz forwomen.

Figure 7 Vorticity and pressure contours at 2 ms intervals.The left column shows vorticity contours, the rightcolumn shows pressure contours. The top rowshows the solution evaluated at t = 6 ms, thesecond row is at t = 8 ms and so on up tot = 14 ms (last row). The colorbar in the vorticitycolumn stretches from 0 to 50 000 s−1 and thecontour levels are spaced 3 750 s−1 apart. For thepressure column, the inflow is at p = p∞ + ∆p, theoutflow is at (approximately) p = p∞ and thecontour levels are spaced 71Pa apart (see onlineversion for colours)

8 Conclusions

Our 2D model for the vocal folds based on thelinear elastic wave equation in the first-order formand the airflow based on the compressible Navier–Stokes equations in the vocal tract proves to be ableto capture the self-sustained pressure-driven oscillationsand vortex generation in the glottis. The high-ordermethod for the linear elastic wave equation with aSAT formulation for the boundary conditions ensuresa time-stable solution. The fluid and structure fieldsare simultaneously integrated explicitly in time andboundary data is exchanged only at the end of a timestep. With this formulation, there is no need for iterationsto find the equilibrium displacement for the structure


depending on the fluid stresses. For the problem weconsider here, the limiting factor on the time step isthe CFL condition from the compressible Navier–Stokesequations. Since the fluid grid of the vocal tract has moregrid points than the structure grids of the vocal foldsand the non-linear flow equations are more involved thanthe linear structure equations, the effort of integratingthe linear elastic wave equation to get the structuredisplacement is small compared with the flow solution.

Acknowledgements

The current research was funded by the SwedishResearch Council under the project ‘NumericalSimulation of Respiratory Flow’.

This is a modified version of our paper with thesame title presented at the 8th International Conferenceon CFD in the Oil and Gas, Metallurgical and ProcessIndustries (Skjetne and Olsen, 2011). More details on theassumptions of our 2D model are added in Section 1Introduction, and Figure 6 is added and discussed inSection 7.4 Numerical simulation.

References

Carpenter, M.H., Gottlieb, D. and Abarbanel, S. (1994) ‘Time-stable boundary conditions for finite-difference schemessolving hyperbolic systems: methodology and applicationto high-order compact schemes’, J. Comput. Phys.,Vol. 111, pp.220–236.

Fornberg, B. (1998) A Practical Guide to PseudospectralMethods, Cambridge University Press, Cambridge.

Gustafsson, B. (2008) High Order Difference Methods forTime-Dependent PDE, Springer-Verlag, Berlin.

Gustafsson, B., Kreiss, H-O. and Oliger, J. (1995) TimeDependent Problems and Difference Methods,John Wiley & Sons, New York.

Hunter, E.J., Titze, I.R. and Alipour, F. (2004) ‘A three-dimensional model of vocal fold abduction/adduction’,J. Acoust. Soc. Am., Vol. 115, No. 4, pp.1747–1759.

Larsson, M. (2007) Numerical Simulation of HumanPhonation, Master Thesis, Department of InformationTechnology, Uppsala University, Sweden.

Larsson, M. (2010) Numerical Modeling of Fluid-StructureInteraction in the Human Larynx, PhD Thesis, NorwegianUniversity of Science and Technology, Trondheim,Norway.

Larsson, M. and Müller, B. (2009a) ‘Numerical simulation ofconfined pulsating jets in human phonation’, Computers& Fluids, Vol. 38, No. 7, pp.1375–1383.

Larsson, M. and Müller, B. (2009b) ‘Numerical simulationof fluid-structure interaction in human phonation’, inSkallerud, B. and Andersson, H.I. (Eds.): MekIT 09 FifthNational Conference on Computational Mechanics, TapirAcademic Press, Trondheim, Norway, pp.261–280.

Larsson, M. and Müller, B. (2010a) ‘High order numericalsimulation of fluid-structure interaction in humanphonation’, Proceedings of ECCOMAS CFD 2010conference, 14–17 June, Lisbon, Portugal.

Larsson, M. and Müller, B. (2010b) ‘Numerical simulationof fluid-structure interaction in human phonation:Application’, in Kreiss, G., Lötstedt, P., Målqvist,A. and Neytcheva, M. (Eds.): Numerical Mathematicsand Advanced Applications 2009 (Proceedings ofENUMATH 2009 Eighth European Conference onNumerical Mathematics and Advanced Applications,29 June–3 July 2009), Uppsala, Sweden, Part 2,Springer-Verlag, Berlin, pp.571–578.

Larsson, M. and Müller, B. (2011) ‘Numerical simulationof fluid-structure interaction in human phonation:Verification of structure part’, in Hesthaven, J.S.and Rønquist, E.M. (Eds.): Spectral and High OrderMethods for Partial Differential Equations (SelectedPapers from the ICOSAHOM ’09 Conference, 22–26June 2009, Trondheim, Norway), Springer-Verlag, Berlin,pp.229–236.

LeVeque, R.J. (2002) Finite Volume Methods for HyperbolicProblems, Cambridge University Press, Cambridge.

Link, G., Kaltenbacher, M., Breuer, M. and Döllinger, M.(2009) ‘A 2D finite-element scheme for fluid-solid-acousticinteractions and its application to human phonation’,Methods in Applied Mechanics and Engineering,Vol. 198, pp.3321–3334.

Luo, H., Mittal, R., Zheng, X., Bielamowicz, S.A., Walsh, R.J.and Hahn, J.K. (2008) ‘An immersed-boundary methodfor flow – structure interaction in biological systems withapplication to phonation’, J. Comput. Phys., Vol. 227,pp.9303–9332.

Müller, B. (1996) Computation of Compressible Low MachNumber Flow, Habilitation thesis, ETH Zurich, Zurich,Switzerland.

Müller, B. (2008) ‘High order numerical simulation of Aeoliantones’, Computers & Fluids, Vol. 37, No. 4, pp.450–462.

Poinsot, T.J. and Lele, S.K. (1992) ‘Boundary conditionsfor direct simulations of compressible viscous flows’,J. Comput. Phys., Vol. 101, pp.104–129.

Sesterhenn, J., Müller, B. and Thomann, H. (1999) ‘On thecancellation problem in calculating compressible low machnumber flows’ J. Comput. Phys., Vol. 151, pp.597–615.

Strand, B. (1994) ‘Summation by parts for finite differenceapproximations for d/dx’, J. Comput. Phys., Vol. 110,pp.47–67.

Titze, I.R. (2000) Principles of Voice Production, NationalCenter for Voice and Speech, Iowa City, USA.

Titze, I.R. (2008) ‘The human instrument’, Scientific American,Vol. 298, pp.94–101.

Visbal, M.R. and Gaitonde, D.V. (2002) ‘On the useof higher-order finite-difference schemes on curvilinearand deforming meshes’, J. Comput. Phys., Vol. 181,pp.155–185.

Zhang, C., Zhao, W., Frankel, S.H. and Mongeau, L. (2002)‘Computational aeroacoustics of phonation, part II’,J. Acoust. Soc. Am., Vol. 112, No. 5, pp.2147–2154.

Zhao, W., Frankel, S.H. and Mongeau, L. (2002)‘Computational aeroacoustics of phonation, part I’,J. Acoust. Soc. Am., Vol. 112, No. 5, pp.2134–2146.


Appendix A

The eigenvector matrix T (kx, ky) of P (kx, ky) reads

T (kx, ky) =

kxcp/λ −ky 0 −ky −kxcp/λ

ky cp/λ kx 0 kx −ky cp/λ

k2xα + k2

y − 2kxky csρ

r2 k2y

2kxky csρ

r2 k2xα + k2

y

2kxkyµ/λ ρcsk −kxky −ρcsk 2µkxky/λ

k2yα + k2

x2kxky csρ

r2 k2x − 2kxky csρ

r2 k2yα + k2

x

(31)

The inverse of this matrix reads

T (kx, ky)−1 =

12r2

kxλcp

kyλ

cp

k2x

αr2 2 kykx

αr2k2

y

αr2

−ky kx − kxky

ρcs

kr2

ρcs

kxky

ρcs

0 0 − 2kα

+4k2

y

βr2 −8 kxky(λ+µ)r2(λ+2µ)

2kα

+ 4 k2x

βr2

−ky kxkxky

ρcs− kr2

ρcs− kxky

ρcs

− λkxcp

− λky

cp

k2x

αr2 2 kykx

αr2k2

y

αr2

(32)

where the parameters are defined by k = (k2x − k2

y)/(k2x +

k2y), r = (k2

x + k2y)1/2, cp = rcp, cs = rcs, α = (λ + 2µ)/λ

and β = αλ/µ.

Nomenclature

Greek symbolsΛ Eigenvalue matrix for linear elastic equationsλ, µ Lamé parameters; λ also refers to an

eigenvalueσ Cauchy stress tensorξ, η Computational coordinates

Latin symbolsA, B Coefficient matrices for the linear elastic

wave equationcp, cs Primary and secondary wave speeds in the

structureF, G Flux vectors in x- and y-directionsf, g, h Components of the Cauchy stress tensor σ in

the structureF c,v, Gc,v Inviscid (c) and viscous (v) flux vectorsgI,II Functions representing the boundary

conditions for characteristic variables uI,II

H Diagonal norm matrix associated withQ

J−1 Jacobian determinant of coordinatetransformation

k = ξ, η Computational coordinatep Pressure in the fluidQ Finite difference operatorq, q Vector of unknowns in the structure

in physical and computationalcoordinates

q Grid function of unknowns in thestructure in computational coordinates

R, L Matrices for right and left goingcharacteristic variables at right and leftboundaries

SAT(k)i Vector value of SAT expression in

characteristic variables at grid point iin direction k = ξ, η

SAT(k)i Vector value of SAT expression in

standard variables at grid point i indirection k = ξ, η

SATi,j Vector value of total SAT expression instandard variables at grid point i, j

T Matrix of eigenvectorst Traction from the fluid at fluid-

structure interfacet TimeU, U Vector of conservative variables in the

fluid in physical and computationalcoordinates

u, v Velocity components in the structureuI,II,uI,II Characteristic variables corresponding

to positive and negative eigenvaluesand their corresponding grid functions

u(k = k0, t),v(k = k0, t)

Specified boundary conditions onvariables u, v at k = k0

U′ Grid function of conservativeperturbation variables in the fluid incomputational coordinates

u Discrete solutionx, y Cartesian coordinates

Sub/superscripts′ Perturbation variablesi, j Indices i, jn Time level n

martin larsson bernhard müller*folk.ntnu.no/bmuller/pcfd_12_2-3_paper10.pdf · reference to this...

Documents